Both to reduce the dimensionality and to provide guidance
as to a possible trend, we do not keep the planet radius as a free parameter
but instead adopt for definiteness the fitsFor convenience, they can be found in a few popular languages in the “Suite of Tools to Model Observations of accRetIng planeTZ” at https://github.com/gabrielastro/St-Moritz.
Radii of up to \(R_{\mathrm {p}}\approx 9\) –12 \(R_{\mathrm {J}}\)  are reached for \(\dot{M} =3\times 10^{-4}~M_{\mathrm {J}}\,\mathrm {yr}^{-1}\) .
This might seem high but is in line with the high-entropy models of [1]} and [2]}. by [3]} of the radius as a function of accretion rate and planet mass in
the population synthesis calculationsSince the planet structure model is the same for gas giants, the newest-generation population synthesis of [4]}, [5]} yields the same results, only with fewer synthetic gas giants and therefore less statistically robust fits. of [6]}, [7]},
for their scenarios of “cold accretion” and of “hot-accretion” (see also [8]}). We call these fits respectively “Cold-population fit” and “Warm-population fit”.
Towards high accretion rates and masses, the radius increases with either quantity, and the warm-population fit is larger than the cold-population fit.
Typical values are \(R_{\mathrm {p}}=1.5\) –3 \(R_{\mathrm {J}}\)  and 2–4 \(R_{\mathrm {J}}\) , with a stronger dependence of \(R_{\mathrm {p}}\)  on \(\dot{M} \)  in the “hot” case.
The “cold-” and “hot accretion” scenarios represent extreme outcomes of the accretion process
in which the entire accretion energy is respectively radiated away at the shock or on the contrary brought into the planet [9]}, [6]}, [11]}, [12]}.
We assume that the radius depends only on the total accretion rate and the mass but not on the filling factor.
